Dear all,
.

`I would like to know the rationale of the asymptotic covariance matrix of
``the Brant test.
``Consider that there is a 3-category dependent variable z. To do Brant test,
``we dichotomise z as two variable, zj, and zl. There two variables are
``separately modelled by two binary logit models, j and l. Let bj is the MLEs
``of the model j, and bj the MLEs of the model l. Brant presented an estimator
``for the covariance between bj and bl (X is the matrix of independent
``variables with a constant vector. Wjl = diag(πl-πjπl), where πj is the
``predicted probability of zj=1 estimated by the model j, and πl is Pr(yl=1)
``estimated by the model l.)
`.
COV(bj,bl) = [(X’ Wjj X)^-1] [X’ Wjl X] [(X’ Wll X)^-1]
.

`Where is this estimator derived from? I know that, if we use this estimator
``to compute cov(bj, bj), then the equation above shrinks back to
`.
COV(bj,bj) = (X’ Wjj X)^-1
.

`which is just the covariance matrix of binary logit model j. However, when
``we compute COV(bj,bl), the estimator looks like a “sandwich”:
`.
COV(bj,bl) = COV(bj,bj) [X’ Wjl X] COV(bl,bl)
.

`The first and the third term in the right-hand side are the covariance
``matrices of the model j and l respectively. But, my questions are: (1) where
``does the middle term in the right-hand side come from? (2) why should the
``three terms in the right-hand side be put together in that way?
`.

`I have compared the covariance computed by Brant’s estimator with that
``computed by the command “suest”. The results were different. So Brant’s
``estimator cannot be explained by the theory of the seemingly unrelated
``estimation. (If they are indeed theoretically different, then which
``estimator is better?) Moreover, I have compared the results computed by
``Brant’s estimator and by gologit2. They are also different. (Specifically,
``I restricted the coefficients of gologit2 to b1 and b2, and then compared
``the e(V) of the gologits with the covariance computed by Brant’s
``estimator.) So Brant’s estimator cannot be directly explained by the way
``that the generalized ordered logit estimates the covariance.
`.

`I’d appreciate it if someone can explain the rationale of Brant’s
``covariance estimator.
``(PS: I know how to compute it. I want to know how it comes from.) Thank you
``very much.
`.
The paper of the Brant test is:

`Brant, Rollin. 1990. "Assessing Proportionality in the Proportional Odds
``Model for Ordinal Logistic Regression." Biometrics 46(4): 1171-1178.
`https://docs.google.com/open?id=0B984NoKuZv46WUdlZlZHbmliV2s
.
.
.

`Chi-lin Tsai
``
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